Acyclic cluster algebras revisited

TL;DR

This paper introduces a new combinatorial approach linking acyclic skew-symmetrizable cluster algebras to hereditary algebra representation theory, providing criteria for c-vectors and simplifying proofs of key properties.

Contribution

It offers a novel combinatorial criterion for c-vectors and a simplified proof of their sign-coherence, connecting cluster algebras with representation theory.

Findings

Established a necessary and sufficient criterion for c-vectors.

Provided a simple proof of sign-coherence of c-vectors.

Enabled elementary derivation of key cluster algebra properties.

Abstract

We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a necessary and sufficient combinatorial criterion for a collection of vectors to be the c-vectors of some cluster in the cluster algebra associated to a given skew-symmetrizable matrix. Our approach also yields a simple proof of the known result that the c-vectors of an acyclic cluster algebra are sign-coherent, from which Nakanishi and Zelevinsky have showed that it is possible to deduce in an elementary way several important facts about cluster algebras (specifically: Conjectures 1.1-1.4 of [Derksen-Weyman-Zelevinsky]).